This report is originally an internal document of YiXin Liu’s group, which is generated at Jan. 25, 2016. The PDF version can be downloaded via the following link. Comments are welcome!
Abstract
This report performs weak inhomogeneity expansion for the continuous Gaussian chain model of a homopolymer in an external field. The aim is to understand where does the Debye function come from. This technique constitutes the main parts of the random phase approximation (RPA). By extending this technique from singlechain formulation to manychain formulation and regarding the potential field arising from interaction with other chains in the same system as an external field, RPA is equivalent to the weak inhomogeneity expansion.
Continuous Gaussian Chain
The normalized single partition function for a continuous Gaussian chain under an external field is given by^{1}
where the physical length is scaled by the radius of gyration of a nonperturb Gaussian chain $R_g=\sqrt{Nb^2/6}$. The volume $V$ is also dimensionless scaled by $R_g^3$.
The propagator $q$ in the above equation satisfies the FokkerPlanck equation, also known as the modified diffusion equation (MDE)
subject to the initial condition
Note that to write the MDE in the form as in eq. \eqref{eq:MDE}, $w(\vx)$ is actually $N$ times the external potential field. Assuming the external field is $W(\vx)$, then $w(\vx)=NW(\vx)$
For a given external field $w(\vx)$, we can obtain the propagator by solving eq. \eqref{eq:MDE}.
Weak Inhomogeneity Expansion
In general, it is impossible to find an analytic solution to the MDE with a general $w(\vx)$. However, a particularly perturbation expansion can be derived when the applied potential field $w(\vx)$ has inhomogeneities that are weak in amplitude. To define such a situation, we introduce the volume average of the potential
and reexpress $w(\vx)$ according to
which serves to define the inhomogeneous part of the field, $\epsilon\omega(\vx)$. For weak inhomogeneities, a small parameter $\epsilon$ ($\abs{\epsilon}\ll 1$) describes their characteristic amplitude. For the continuous Gaussian chain model of a homopolymer, the MDE and initial condition become
where the functional dependence of q on $w(\vx)$ has been suppressed in our notation. The term proportional to $w_0$ on the righthand side of eq.\eqref{eq:MDEw0} can be removed by the substitution
which leads to
A weak inhomogeneity expansion can be developed by assuming that $p(\vx, s)$ can be expressed as
where the $p^{(j)}(\vx, s)$ are independent of $\epsilon$. In eq.\eqref{eq:pexpansion} we adopt the conventional notation $\sim$ to indicate an asymptotic expansion. As such, the infinite series on the righthand side may be either convergent or divergent. Even when it does not converge, eq.\eqref{eq:pexpansion} can still be useful in truncated form for approximating $p(\vx, s)$ at sufficiently small $\epsilon$.
The $p^{(j)}$ are calculated by inserting eq.\eqref{eq:pexpansion} into eq.\eqref{eq:MDEomega} and equating terms order by order in $\epsilon$.
Zeroth Order Solution
At leading order, $O(\epsilon^0)$, we have
which has the trivial solution
First Order Solution
At $O(\epsilon^1)$, the corresponding equations (see Appendix A for derivation) are
Provided the system under consideration is unbounded or subject to periodic boundary conditions, this initial value problem is most easily solved by means of spatial Fourier transforms. Defining Fourier transforms in accordance with
and assuming that the Fourier transform of $\omega(\vx)$ exists, denoted by $\hat{\omega}(\vk)$, one finds that (See Appendix B for derivation)
where the carets denote Fouriertransformed quantities and
Second Order Solution
At $O(\epsilon^2)$, the corresponding equations (see Appendix C for derivation) are
A similar procedure leads to (see Appendix D for derivation)
with
Expansion of $Q$
We can expand the normalized single partition function $Q$ based on above perturbation expansion of the propagator $q$. The propagator being expanded to $O(\epsilon^2)$ is
Therefore we can expand $Q$ as
It is more convenient to write the integrals of $p^{(j)}$ in Fourier space, which can be expressed as
and
Now we can express $Q$ in the Fourier space as
The expansion of $Q$ can be further simplified. Firstly, we know that the volume average of the fluctuation of the potential field is 0 because
The value of the Fourier transform of the potential field at zero wave number is equal to this average because
Therefore, one obtains
Substituting this into eq.\eqref{eq:p1} gives
Secondly, from eq.\eqref{eq:p2} we know
In the second line, we only change the summation variable from $\vk’$ to $\vk$. Noting that
Let
Then it becomes
This is where the well known Debye function comes from, which is defined as
Therefore,
Insert eq.\eqref{eq:p101Fourier}, \eqref{eq:p201h3}, and \eqref{eq:h301final} into eq.\eqref{eq:Qexpanv2}, we arrive the final expansion of $Q$ in the Fourier space
or, by inverting the Fourier transforms,
If we define the inverse transform of the Debye function as
Then the expansion of $Q$ in real space can be written as
Expansion of the Density Operator
A weak inhomogeneity expansion for the segment density operator $\rho(\vx; [w])$ can be obtained in one of two equivalent ways. One way is expressing $\rho(\vx; [w])$ as an integral of the propagator, such as
and substituting the weak inhomogeneity expansion of the propagator eq.\eqref{eq:qp} and \eqref{eq:pexpansion} into above equation and keeping to a certain order. The other way is performing a direct functional differentiation of eq.\eqref{eq:Qexpanfinal} according to
Here we Follow the latter approach. From eq.\eqref{eq:Qexpanfinal} we have
From eq.\eqref{eq:w0w} we have
From eq.\eqref{eq:womega} we have
Now we can perform the functional differentiation as
where $\rho_0 \equiv N/V$ is the volumeaverage segment density of a single chain. If we only retain the first order contribution of $\epsilon$, above equation can be simplified to
Or we can use eq.\eqref{eq:womega} to rewrite $\rho$ as a functional of $w$ instead of $\omega$
We can also inverse the above equation to predict what external field should be applied if we want to obtain a certain segment density. Such inversion can be done in the Fourier space
where $\phi=\rho/\rho_0  1$ is the dimensionless fluctuation of segment density around its volumeaverage value and $\Delta w=ww_0$ is the fluctuation of the external field. It is straightforward to inverse it
Appendix
A. Derivation of eq.\eqref{eq:MDEp1}
At $O(\epsilon^1)$, the expansion for $p(\vx, s)$ is
Inserting it into eq.\eqref{eq:MDEomega}, we have
From the zeroth order expansion eq.\eqref{eq:MDEp0}, we know
It simplifies eq.\eqref{eq:MDEfullA} to
As $\epsilon \to 0$, the last term in the righthand side of the above equation vanishes, leading to eq.\eqref{eq:MDEp1}.
B. Derivation of eq.\eqref{eq:p1}
We know
which is the trivial solution of zeroth order expansion. Therefore eq.\eqref{eq:MDEp1} becomes
Perform Fourier transforms on both sides of above equation, we reaches
Reorganize above equation into the form
It can be easily solved and the solution is
Let $s=0$, and with $\hat{p}^{(1)}(\vk, 0)=0$ (because $p^{(1)}(\vx, 0)=0$), we then have
or
Write eq.\eqref{eq:p1intermediateA} with $\hat{p}^{(1)}$ in the lefthand side and substitute above equation into it, we arrives at
which is equivalent to eq.\eqref{eq:p1}.
C. Derivation of eq.\eqref{eq:MDEp2}
At $O(\epsilon^2)$, the expansion for $p(\vx, s)$ is
Inserting it into eq.\eqref{eq:MDEomega}, we have
Substitute eq.\eqref{eq:MDEp0} and \eqref{eq:MDEp1} into above equation and ignore terms with $\epsilon$, it simplifies to eq.\eqref{eq:MDEp2}.
D. Derivation of eq.\eqref{eq:p2}
We copy eq.\eqref{eq:MDEp2} here
Perform Fourier transforms on both sides of above equation, we arrives
This is a general first order linear partial differential equation with respect to $s$
The solution is
where $C$ is a constant which can be obtained by applying the initial condition, and
From the first order solution, we can find
Substituting eq.\eqref{eq:p1} into above equation, we have
To simplify the notation, we define
Substitute eq.\eqref{eq:usA} and \eqref{eq:wp1A} into eq.\eqref{eq:p2rawA}
Because $p^{(2)}(\vx, 0)=0$, thus $\hat{p}^{(2)}(\vk, 0)=0$, therefore
Thus
Substitute it back into previous equation,
We can define
which completes the derivation.
Acknowledgements
This note is supported by the by the China Scholarship Council (No. 201406105018).
References

Fredrickson, G. H. The Equilibrium Theory of Inhomogeneous Polymers; Clarendon Press: Oxford, 2006. ↩